Vortical Flows

Cylindrical Bubble Dynamics

Cavitation induced "singing" vortex.
See and hear singing vortex (2.3MB MPEG)
See in slow motion (0.4MB MPEG)

A relatively high amplitude, discrete tone is radiated from fully developed tip vortex cavitation under certain conditions. The phenomenon of the "singing vortex" was first reported by Higuchi et al (1989). Since that time we have more closely examined the singing phenomenon by varying the hydrofoil cross-section, scale, angle of attack, water quality and cavitation number. Noise data were collected for each condition with visual documentation using both still photography and high speed video in an effort to explain the mechanism of vortex singing.

Cavitation testing and force measurements were made in two water tunnels, one at SAFL which has a 190 mm square cross section (Arndt, et al 1991) and the other at Obernach, Germany with a 300 mm square cross section (Arndt & Keller, 1992). Oil flow visualizations were obtained in two wind tunnels, one at the Department of Aerospace Engineering at the University of Minnesota (Higuchi et al, 1987) and the second at SAFL (originally an air model of the HYKAT facility, Wetzel & Arndt, 1994).

Observations of cavitation were made with either conventional still photography or with high speed video. A new Kodak video camera with the possibility of framing rates as high as 40,500 fps was used at Obernach in conjunction with still photography at SAFL, also using a standard Nikon camera. Video observations were made at a framing rate of 4500 fps, which was more than adequate for observing phenomena that had a frequency less than 600 Hz. Data were collected over a range of lift coefficients, velocities and water quality in both facilities.

Radiated sound was measured in the SAFL tunnel with an hydrophone positioned above the hydrofoil tip in a tank of quiescent water that was separated from the test section by a thin plate of plexiglass (Higuchi et al, 1989). The hydrofoil was mounted at the floor of the test section and the thin plexiglass plate formed the roof of the test section. A similar setup was used in Obernach, except that a single hydrophone was mounted in a tank of water that was positioned against one of the side windows of the test section. Therefore the observation angle differed by approximately 90o in the two test facilities. The significant differences in the acoustic path for measurements in the two facilities precluded comparison of amplitude data. Only frequency data were compared.

The conditions at which singing is observed to occur are limited. For a given free-stream velocity and angle of attack, singing only occurs within a narrow band of cavitation number, Δσ = ±0.15 from the test condition, σ_s. While varying through this narrow band, the frequency of the tone is observed to decrease with decreasing sigma. An example of the singing phenomenon is to the left. This video is taken at a velocity of 10 m/s at normal framing rate. The second video below it is made by photographing the event at a framing rate of 4500 fps and playing back at 25 fps, thus providing a magnification in time of 180.

The high speed video images provide a convenient method to verify the frequencies measured with the hydrophones and confirm that the noise is produced by undulations of the vortex core. A plot of frequency measured from the video images versus that obtained with the hydrophones is shown in Figure 1. It is important to note that the core radius, a, and the wave length, λ, can also be measured.

Figure 1. Comparison of hydrophone and observed frequencies. ENLARGE

The experimental observations suggest a standing wave on the surface of the hollow vortex core. Kelvin (1880) studied the wave pattern on a stationary, irrotational hollow core vortex. He found two dominate helical modes, one rotating with the same sense as the vortex and the other rotating and propagating in the opposite direction. The rotational speeds, 2πf, are given by Equation 1, where is the rotational speed of the vortex and N is a function of wave number based on core radius, ka = a/λ, and is numerically greater than unity.

A standing wave is possible when the vortex is superimposed on a uniform axial flow. This can occur when the celerity of the counter-rotating mode is equal and opposite to the free stream velocity, U (negative frequency in equation 1). This occurs when

This suggests the surprising result that a standing wave will occur at only a single value of . This finding is in qualitative agreement with the observation that singing only occurs over a very narrow range of cavitation number. This result is also qualitatively consistent with the observation that singing only occurs over a very narrow range of cavitation number.

Although this analysis provides a framework for reducing the data, it is qualitative at best. Using these ideas we are able to correlate all the data obtained in both facilities in the form

The strongest tone occurred at 2πfa/U = ka = 0.5 and σ_s = 1.2. Noise amplitude did vary for each data point, but could not be accurately measured in this frequency range. Had it been possible to accurately measure amplitude, it may have been possible to fit the data in Figure 2 with iso-contours of amplitude that would be elliptical in shape with their major axes aligned at a slope of 0.45.

Figure 2. Comparison Strouhal number and cavitaion number. ENLARGE

Throughout the test program singing was only observed when the hollow core was attached to the tip. Thus it is possible that the mechanism for singing is also associated with a complex interaction between the tip boundary layer and the attached cavity. High speed video observations indicate that the attached tip cavity oscillates with the same frequency as the pulsating vortex. Figure 3 highlights the relationship between the tip cavity and the average boundary layer characteristics. This picture was created by superimposing a single image of a tip cavity at a cavitation number slightly higher than σs with a photograph of oil film streaklines taken at an equivalent condition in a wind tunnel. Note that the extent of the cavity correlates well with separation of the boundary layer in the tip region at this phase of the cycle.

Figure 3. Close-up of the tip cavity.

References